Let K be a simplicial complex and g the rank of its p-th homology group H p (K) defined with Z 2 coefficients. We show that we can compute a basis H of H p (K) and annotate each p-simplex of K with a binary vector of length g with the following property: the annotations, summed over all p-simplices in any p-cycle z, provide the coordinate vector of the homology class [z] in the basis H. The basis and the annotations for all simplices can be computed in O(n ω ) time, where n is the size of K and ω < 2.376 is a quantity so that two n × n matrices can be multiplied in O(n ω ) time. The pre-computation of annotations permits answering queries about the independence or the triviality of p-cycles efficiently.

The fractures of thin plates often exhibit complex physical behaviors in the real world. In particular, fractures caused by tearing are different from fractures caused by in-plane motions. In this paper, we study how to make thin-plate fracture animations more realistic from three perspectives. We propose a stress relaxation method, which is applied to avoid shattering artifacts after generating each fracture cut. We formulate a fracture-aware remeshing scheme based on constrained Delaunay triangulation, to adaptively provide more fracture details. Finally, we use our multi-layered model to simulate complex fracture behaviors across thin layers. Our experiment shows that the system can efficiently and realistically simulate the fractures of multi-layered thin plates.

As a direct consequence of software quirks, designer errors, and representation flaws, often three-dimensional shapes are stored in formats that introduce inconsistencies such as small gaps and overlaps between surface patches. We present a new algorithm that simultaneously repairs imperfect geometry and topology while generating Delaunay meshes of these shapes. At the core of this approach is a meshing algorithm for input shapes that are piecewise smooth complexes (PSCs), a collection of smooth surface patches meeting at curves non-smoothly or in non-manifold configurations. Guided by a user tolerance parameter, we automatically merge nearby components while building a Delaunay mesh that has many of these errors fixed. Experimental evidence is provided to show the results of our algorithm on common computer-aided design (CAD) formats. Our algorithm may also be used to simplify shapes by removing small features which would require an excessive number of elements to preserve them in the output mesh.

Figure 1: Coke foam. By representing foam geometry using a weighted Voronoi diagram, our particle-based algorithm can efficiently provide bubble features in existing liquid animation. This example contains up to 100K bubbles and each frame takes less than 20 seconds to simulate. Abstract Bubbles and foams are important features of liquid surface phenomena , but they are difficult to animate due to their thin films and complex interactions in the real world. In particular, small bubbles (having diameter <1cm) in a dense foam are highly affected by surface tension, so their shapes are much less deformable compared with larger bubbles. Under this small bubble assumption, we propose a more accurate and efficient particle-based algorithm to simulate bubble dynamics and interactions. The key component of this algorithm is an approximation of foam geometry, by treating bubble particles as the sites of a weighted Voronoi diagram. The connectivity information provided by the Voronoi diagram allows us to accurately model various interaction effects among bubbles. Using Voronoi cells and weights, we can also explicitly address the volume loss issue in foam simulation, which is a common problem in previous approaches. Under this framework, we present a set of bubble interaction forces to handle miscellaneous foam behaviors , including foam structure under Plateau's laws, clusters formed by liquid surface bubbles, bubble-liquid and bubble-solid coupling, bursting and coalescing. Our experiment shows that this method can be straightforwardly incorporated into existing liquid simula-tors, and it can efficiently generate realistic foam animations, some of which have never been produced in graphics before.

Figure 1: Coke foam. By representing foam geometry using a weighted Voronoi diagram, our particle-based algorithm can efficiently provide bubble features in existing liquid animation. This example contains up to 100K bubbles and each frame takes less than 20 seconds to simulate. AbstractBubbles and foams are important features of liquid surface phenomena, but they are difficult to animate due to their thin films and complex interactions in the real world. In particular, small bubbles (having diameter <1cm) in a dense foam are highly affected by surface tension, so their shapes are much less deformable compared with larger bubbles. Under this small bubble assumption, we propose a more accurate and efficient particle-based algorithm to simulate bubble dynamics and interactions. The key component of this algorithm is an approximation of foam geometry, by treating bubble particles as the sites of a weighted Voronoi diagram. The connectivity information provided by the Voronoi diagram allows us to accurately model various interaction effects among bubbles. Using Voronoi cells and weights, we can also explicitly address the volume loss issue in foam simulation, which is a common problem in previous approaches. Under this framework, we present a set of bubble interaction forces to handle miscellaneous foam behaviors, including foam structure under Plateau's laws, clusters formed by liquid surface bubbles, bubble-liquid and bubble-solid coupling, bursting and coalescing. Our experiment shows that this method can be straightforwardly incorporated into existing liquid simulators, and it can efficiently generate realistic foam animations, some of which have never been produced in graphics before.